A private person's arrest (aka a citizen's arrest) must be supported by probable cause - even if said cause is merely the statement of the alleged victim. And, no, the officers did have a choice - they could have taken you to jail (where you would have been released with a notice to appear ... i.e. a ticket). They opted to cite you at the scene, likely because the evidence was weak, and the chances of the altercation continuing was small to non-existant.
As for reading one's rights, this is not required for an arrest, only for an interrogation while in custody. Very few arrests result in Miranda rights being read.
Unless you receive word IN WRITING that you no longer have to appear at the location indicated on the cite at the listed date and time, plan to attend. While the victim may have asked for the matter to be dropped - and it may have been - unless you receive written confirmation, go to court as indicated!

